In this paper, we investigate the fundamental multiple access (MA) scheme in an active intelligent reflecting surface (IRS) aided energy-constrained Internet-of-Things (IoT) system, where an active IRS is deployed to assist the uplink transmission from multiple IoT devices to an access point (AP). Our goal is to maximize the sum throughput by optimizing the IRS beamforming vectors across time and resource allocation. To this end, we first study two typical active IRS aided MA schemes, namely time division multiple access (TDMA) and non-orthogonal multiple access (NOMA), by analytically comparing their achievable sum throughput and proposing corresponding algorithms. Interestingly, we prove that given only one available IRS beamforming vector, the NOMA-based scheme generally achieves a larger throughput than the TDMA-based scheme, whereas the latter can potentially outperform the former if multiple IRS beamforming vectors are available to harness the favorable time selectivity of the IRS. To strike a flexible balance between the system performance and the associated signaling overhead incurred by more IRS beamforming vectors, we then propose a general hybrid TDMA-NOMA scheme with device grouping, where the devices in the same group transmit simultaneously via NOMA while devices in different groups occupy orthogonal time slots. By controlling the number of groups, the hybrid TDMA-NOMA scheme is applicable for any given number of IRS beamforming vectors available. Despite of the non-convexity of the considered optimization problem, we propose an efficient algorithm based on alternating optimization, where each subproblem is solved optimally. Simulation results illustrate the practical superiorities of the active IRS over the passive IRS in terms of the coverage extension and supporting multiple energy-limited devices, and demonstrate the effectiveness of our proposed hybrid MA scheme for flexibly balancing the performance-cost tradeoff.